Mathematical Structure and The Structure of Mathematics Promoting Mathematical Thinking Mathematical Structure and The Structure of Mathematics Anne Watson & John Mason NZAMT July 2015 The Open University Maths Dept University of Oxford Dept of Education

Outline A familiar and pervasive structure A pervasive structure Extending the domain of action A pervasive structure Extending the domain of action (implied) Structuring something less familiar Something new to explore

What does addition mean if you add 27 to 48 using teddies?

What does addition mean if you add 27 to 48 with place value counters What does addition mean if you add 27 to 48 with place value counters ... or coins?

What does addition mean if you add 27 to 48 using Cuisenaire rods?

What does addition mean if you add 27 to 48 using liquid measure?

What does addition mean if you add 27 to 48 using the steel measure?

What does addition mean if you add 27 to 48 using the hundred square?

(and other grids)? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

What does addition mean if you add 27 to 48 using tape?

What does addition mean if you add 27 to 48 using squared paper?

Expressing the structure of addition a + b = c c = a + b b + a = c c = b + a c – a = b b = c - a c – b = a a = c - b

Extending the meaning of addition What can addition mean if you add 27 to 48 using area under y = 1? What can addition mean if you add 27 to 48 using area under y = 3? What can addition mean if you spot that 27 and 48 have common factors and re-write it as 3(9 + 16)? What can addition mean if you add 27 to 48 using area under y = 2x?

Difference Write down two numbers/lengths/quantities with a difference of 3 … and two more numbers with a difference of 3 … and another very different pair Write down two quadratics whose roots differ by 3 ... Write down two definite integrals on the same interval that differ by 3 On consecutive intervals of the same length …

Reprise Enactive experiences towards an appreciation of addition and building of iconic images Symbolic generalisation of additive relationships (structure) Extending to new contexts Focus on some feature (difference)

Multiplicative structure a = bc bc = a a = cb cb = a b = a c = b a c c = a b = c a b

Questions about multiplicative structure How many …. in ….? How many times ….? How much is left over when … ? How many times bigger/smaller … ?

Ratio Write down two numbers/lengths/quantities with a ratio of 3 : 4 … and two more numbers with a ratio of 3 : 4 … and another very different pair Write down two measurements in the ratio 3 : 4 … and another Draw a rectangle whose sides are in the ratio 3 : 4

Reprise Enactive experiences towards an appreciation of multiplication (as repetition and as scaling) and building of iconic images Symbolic generalisation of multiplicative relationships (structure) Extending to new contexts (implied) Focus on some feature (ratio)

LCM & GCD What is the LCM of 27 and 48? What is the LCM of two numbers? What is the GCD (HCF) of 27 and 48? What is the GCD (HCF) of two numbers? The largest number that divides exactly into both numbers The smallest number exactly divisible by both numbers

LCM & GCD What is the LCM of 27/14 and 48/35? What is the GCD (HCF) of 27/14 and 48/35? The smallest fraction exactly divisible by both numbers The largest fraction that divides exactly into both numbers Want these to be integers Want these to be integers So x has to divide both 27 and 48 & y has to be divisible by both 14 and 35 So w has to be divisible by both 27 and 48 & z has to divide into both 14 and 35

What is the period? Period 1 Period 2 Sin(2πx) Sin(πx) Period 3/2

Combined Periods Period 2 Sin(πx) Period 3 Sin(2πx/3) Sin(πx)+Sin(2πx/3) Period 6

Construct your own sine waves with specified periods Period Rehearsal sin(2πx)? Period 1 Period 1/2 sin(6πx)? Period 1/3 Sin(4πx) Sin(10πx)? Period 1/5 Sin(10πx/3)? Period 3/5 Construct your own sine waves with specified periods

Two Fractional periods

Reprise A familiar and pervasive structure (addition) Extending the domain of action A pervasive structure (multiplication) Extending the domain of action (implied) Structuring something less familiar (lcm, gcd, periodicity) Something new to explore (periodicity)

Follow Up Anne.Watson@education.ox.ac.uk John.Mason@open.ac.uk Mathematics as a Constructive Activity: learner generated examples (Erlbaum) PMTheta.com for applets, PPTs, and more